L'Hôpital's rule - Knowlpedia

L'Hôpital's rule (0/0 form, one-sided): Let $f$ and $g$ be continuous on $[a,b)$ and differentiable on $(a,b)$ , and assume that $g'(x)\neq 0$ for all $x\in(a,b)$ . Suppose

\lim_{x\to a^+} f(x)=0 \quad\text{and}\quad \lim_{x\to a^+} g(x)=0,

and that $g(x)\neq 0$ for all $x$ sufficiently close to $a$ with $x>a$ . If the limit

L=\lim_{x\to a^+}\frac{f'(x)}{g'(x)}

exists (as a finite number or as $\pm\infty$ ), then the limit

\lim_{x\to a^+}\frac{f(x)}{g(x)}

also exists and equals $L$ .

Remarks

Analogous statements hold for left-hand limits and for the $\infty/\infty$ indeterminate form, and there are versions for limits at infinity. The proof is based on the Cauchy mean value theorem and is formulated in terms of one-sided limits.